Convex functions and their relatives are ubiquitous in a large variety of applications such as optimization theory, mass transportation, mathematical economics and geometric inequalities related to isoperimetric problems. This chapter is devoted to a succinct presentation of their theory in the context of real normed linear spaces, but most of the illustrations will refer to the Euclidean space \(\mathbb {R}^{N},\) the matrix space \(\operatorname {M}_{N}(\mathbb {R})\) of all \(N\times N\) -dimensional real matrices (endowed with the Hilbert–Schmidt norm or with the operator norm) and to the Lebesgue spaces \(L^{p}\left( \mathbb {R}^{N}\right) \) with \(p\in [1,\infty ]\) .